Phase space renormalization and finite BMS charges in six dimensions

We perform a complete and systematic analysis of the solution space of six-dimensional Einstein gravity. We show that a particular subclass of solutions — those that are analytic near I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{I} $$\end{document}+ — admit a non-trivial action of the generalised Bondi-Metzner-van der Burg-Sachs (GBMS) group which contains infinite-dimensional supertranslations and superrotations. The latter consists of all smooth volume-preserving Diff×Weyl transformations of the celestial S4. Using the covariant phase space formalism and a new technique which we develop in this paper (phase space renormalization), we are able to renormalize the symplectic potential using counterterms which are local and covariant. The Hamiltonian charges corresponding to GBMS diffeomorphisms are non-integrable. We show that the integrable part of these charges faithfully represent the GBMS algebra and in doing so, settle a long-standing open question regarding the existence of infinite-dimensional asymptotic symmetries in higher even dimensional non-linear gravity. Finally, we show that the semi-classical Ward identities for supertranslations and superrotations are precisely the leading and subleading soft-graviton theorems respectively.


Introduction
The asymptotic symmetry group (ASG) of a geometry pM, Gq is defined as the set of all transformations which preserve the boundary and gauge conditions that define the metric G and which act non-trivially on the asymptotic boundary data. This is often described by a simple equation

ASG "
Allowed Transformations Trivial Transformations . (1.1) For four-dimensional asymptotically flat spacetimes, the asymptotic symmetry group was first studied by Bondi, Metzner, van der Burg [1] and Sachs [2] in the early 60s. A surprising outcome of their analysis was that the ASG for asymptotically flat spacetimes is an infinite-dimensional extension of the Poincaré group which today we refer to as the BMS group. This group consists of the usual 4D Lorentz transformations and an infinite-dimensional extension of translations known as supertranslations (no relation to supersymmetry). This group has received much attention over the past decade due to the seminal work by Strominger [3] who proved that the semi-classical Ward identity for supertranslation symmetry [4] is the leading soft-graviton theorem [5]. It has also been suggested [6][7][8] that the analysis of BMS can be generalised to include an infinite-dimensional generalisation of Lorentz transformations, aptly named superrotations. Superrotations generalise Lorentz transformations -which are isomorphic to global conformal transformations of the celestial S 2 (i.e. the sphere at infinity) -to the local Virasoro transformations. It was shown in [9] that the Ward identity of this symmetry is a new, hitherto unknown subleading soft-graviton theorem [10].
These asymptotic symmetries, the corresponding soft theorems together with the associated gravitational memory effects [11,12], form what is now known as infrared triangles (see [13] for an early review). These triangles -which are now known to be ubiquitously present in many theories in asymptotically flat spacetimes -thread together three previously unrelated fields of research -formal relativity (asymptotic symmetries), perturbative QFT (infrared effects) and observational/experimental relativity (memory effects). This remarkable discovery has unveiled many interesting features of gauge and gravitational theories and has revitalised research in flat holography. In particular, the infrared triangle is a universal feature that any microscopic/holographic formulation of quantum gravity in asymptotically flat spacetimes will have to exhibit, independently of the fine-grained details of the model. Indeed, over the past decade, two separate (but perhaps, equivalent) formulations of flat holography have emerged -Celestial Holography and Carrollian Holography [14][15][16][17] both of which manifest the infrared triangle in slightly different ways! Exploration of the rich structure described above has largely been limited to four spacetime dimensions. Efforts to analyse and study this structure in dimensions greater than four are sparse [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. There are several reasons for this. A BMS-like analysis reveals that the asymptotic symmetry group of asymptotically flat gravity is simply the finite-dimensional Poincaré group [19], i.e.
there is no infinite-dimensional extension! It is also well-established that while four-dimensional theories with massless particles (e.g. graviton) are plagued with infrared divergences, there are no such divergences in higher dimensions. Finally, it was shown in [34] that gravitational memory effects are also not present in higher dimensions. Altogether, these results seem to suggest something quite obvious -namely that the infrared sector of higher dimensional gravity is trivial. There is simply not enough structure to admit the richness we see in four dimensions!
The gaping outliers in all of this are the soft theorems. The leading and subleading soft-graviton theorems exist and have the exact same qualitative structure in all dimensions [35]. This fact alone has forced us and others to revisit the aforementioned results and to question if it is indeed true that BMS symmetries are absent in higher dimensions. The first of this "new-wave" of results was obtained in [22] where it was shown that by weakening the definition of asymptotic flatness employed in [19], the asymptotic symmetry group in higher even dimensions can include supertranslations.
Working in linearised gravity, it was then shown these supertranslations are related to the leading soft-graviton theorem, exactly as in four dimensions! Following this, [24] showed that there are memory effects in higher dimensions, albeit at a subleading order (which is why they were not observed in [34]) and that they are related to supertranslations and soft theorems in the usual way. These results were partially generalised to non-linear gravity in [32]. Despite the progress, no coherent analysis of the inevitable divergences of such supertranslation-compatible phase spaces has yet been given.
Having gathered enough evidence about the existence of supertranslations in higher dimensions, the obvious next step is to question the existence of superrotations. As mentioned before, the subleading soft theorem exists in all dimensions. Does this not imply that superrotations should also exist in all dimensions? At this stage, we encounter our first complication. Superrotations in four dimensions were obtained as a Virasoro generalization of the global conformal transformations, SLp2, Cq Ñ VirˆVir. Such an extension can be realised on the S 2 because of the holomorphicity of two-dimensional conformal transformations. In higher dimensions, we lose this holomorphic structure and there is no "local conformal group". The conformal group on S d is the finite-dimensional group SOp1, d`1q which cannot reproduce the infinitely-many subleading soft theorems (one for each point on S d ).
This problem actually has a relatively straightforward resolution. To understand this, we revisit the four-dimensional problem. One of the issues with the Virasoro superrotations is that they are only defined locally on the celestial sphere. The Hamiltonian charges that generate superrotation transformations on the phase space involve integrals over the entire S 2 and are therefore ill-defined.
This issue was observed by Campiglia and Laddha in [36] and they proposed an alternative definition of superrotations which are parameterized by smooth vector fields on S 2 . In doing so, they resolved some of the technical issues that lie at the heart of the equivalence between soft theorems and asymptotic symmetries while still preserving the infrared triangle. For our purposes, what is particularly interesting about this definition of superrotations is that it naturally generalises to higher dimensions! This natural expectation prompted the analyses of [30,33,37,38]. Specifically, closely following the linearised gravity analysis of [22], the early work [37] concocts the Poisson bracket of a pair of asymptotic fields and partially derives the subleading soft theorem. The work [38] explores superrotation charges within the context of linear higher-spin theories. Later, in full non-linear general relativity, the authors of [33], adapting [32] and [30] (about which we will say more shortly), provide a partial construction of superrotation charges linearising around a BMS frame and the matching with the subleading soft theorem in the case of scalar external particles.
Having identified the right set of transformations that we are after, we must next understand what boundary conditions one should use to define asymptotic flatness such that these transformations would be allowed, but not trivial. The goal of this paper is to answer this question and more generally, to formalize the rather scattered discussion of the BMS group in higher dimensions. In the process of formalization, we shall settle several questions whose answers are either completely unknown or only partially known, e.g. 7. How are these symmetries related to the leading and subleading soft-graviton theorems?
In this paper, we will answer every single question listed above and provide "first-principles" derivations of all our results. The starting point is the general analysis of boundary conditions in dimensions greater or equal to four performed in [30]. Along the way, we will encounter many of the usual complications associated with superrotations and several new ones that are unique to higher dimensions. The first, and most obvious complication is the dramatic increase in the difficulty of calculations. To simplify our work, we restrict ourselves to six spacetime dimensions. The results of this paper can be easily generalised to all higher even dimensions, although the explicit calculations are more and more tedious as we go higher up in dimension. 1 To facilitate our calculations, we have extensively used Mathematica and the package diffgeo.m. 2 The second, far more important, complication that we encounter are the large volume divergences which arise in the construction of the generalised phase space. To deal with this issue, we develop a novel formalism which we refer to as "phase space renormalization" which allows us to regulate and renormalize all the divergences in a local and covariant way. Once this hurdle is crossed and we have a finite symplectic potential, the rest of the construction follows in a relatively straightforward manner.
We conclude the introduction with a brief comment on the philosophy that has guided our work, namely AdS/CFT. There, in order not to over-constrain the CFT and to properly develop the holographic dictionary via holographic renormalization [39,40], the gravitational solution with the most general boundary conditions is first constructed. Any additional restriction on the bulk solution often implies extra constraints on the CFT side which changes the physics of the theory.
In the same vein, we proceed in this paper by first studying the most general solution space that is consistent with Einstein's equations and imposing additional restrictions only when absolutely necessary! Another point where the AdS/CFT line of reasoning plays a role is our approach toward the renormalization of the phase space. We insist on finding local and covariant counterterms, as is done in the holographic renormalization of the bulk AdS action. However, in contrast to AdS/CFT, we do not attempt to renormalize the action. Instead, we only consider a renormalization of the phase space. 3

Outline of Paper and Results
Since this paper is rather lengthy and the discussions across sections are intertwined, we present in this section a detailed outline of the paper along with the important results.
Constructive Definition of Analytic Solutions from Boundary Conditions at IẀ e begin this paper with Section 2, which describes three different classes of solutions of Ricci-flat gravity in dimensions greater than four.
In Section 2.2.1, we present the results from the previous work of one of us [30]. The metric is given in Bondi-Sachs gauge (this gauge is used throughout this paper as well), where every component is a function of all the coordinates and is asymptotically expanded in terms of the radial coordinate r. The procedure put forward in [30] shows that generic solutions are polyhomogeneous, containing logarithmic terms in the expansion, and allows to identify all such independent terms and their descendants. The qualitative structure is presented in the main text and the explicit (and gory) details of the solutions are in Appendix B.1.
In Section 2.2.2, we construct a smaller solution space by imposing the vanishing of the Riemann tensor at I`in an orthonormal frame This condition, along with an appropriate definition of the frame fields, reduces the general solution space to an analytic solution space S analytic without logarithmic terms in the radial expansion (this is proved in Appendix B.2). This is the solution space that we focus on in the majority of the paper.
An important physical implication of this condition is that gravitational radiation vanishes in the far past of I`, i.e. near I`(see Section 2.5 for discussion). The specific fall-off near I`is derived in Section 2.4.1 using a linear approximation around Minkowski spacetime (this is done in Section 2.3) and is further supported in Section 2.4.2 by the requirement that the large u fall-offs must be consistent with the action of the subsequently derived diffeomorphism group.
Metrics in S analytic are characterized by the following data on I`-radiation r h (2)AB pu, xq, mass aspect M pxq, angular momentum aspect N A pxq, four-dimensional diffeomorphisms χ A pxq, Weyl factor Φpxq, and a scalar function Cpxq: Firstly, it highlights the appropriate variables to work with, as it clearly differentiates between canonical degrees of freedom and large gauge degrees of freedom. Furthermore, it helps in treating the divergences in the pre-symplectic potential, as we will discuss in the next paragraph.
All Divergences in the Pre-Symplectic Potential are Boundary Terms The immediate issue one faces when trying to construct the charges associated to the Weyl BMS diffeomorphism group is that the symplectic form computed on S analytic has large r divergences. To proceed, we must then regulate and renormalize all these divergences.
The first step towards this goal is the realization that all large r divergences of the pre-symplectic potential, for a metric in S analytic , reduce to a boundary term. The way to prove this is by using (1.4), which implies a splitting of the pre-symplectic potential in the following way where r‹θ can pδqs µ " G µν B νX ρ´r ‹θpδqs ρ | XÑX¯, r‹Q ξ s µν " 2∇ rµ ξ νs . (1.6) Given thatθpδq is the symplectic potential density for the canonical metricG, it is straightforward to see that the bulk part θ can encodes the variation of the canonical degrees of freedom, and the boundary part Q { δX captures the variations of the large gauge fields -δC, δχ A and δΦ. In Section 4.1, we show that the bulk part is finite and that all the divergences are in the boundary contribution which is derived in Section 4.2. This result relies crucially on the boundary conditions we chose, highlighting again their importance. It is not clear whether relaxing them would preserve it.

Renormalization via Local and Covariant Counterterms
At this point, anyone who is familiar with covariant phase space formalism can think that the divergence issue is immediately resolved following [43,44]. The symplectic potential density is a pd´1q-form, whose definition is ambiguous, up to the addition of an exact form (or in other words a boundary term). Using this fact, such divergences can be naively cancelled by simply adding to θ the divergent pieces with opposite sign (see the discussion around (4.29)). This is however not enough to provide a sensible notion of renormalization as it appears to us to be too arbitrary.
Without any sort of first principles prescription for the renormalization, pieces can be added at will to θ at any order and hence also the finite part can be rendered highly ambiguous, if not cancelled altogether! In this paper, we propose a prescription for local and covariant counterterms (in an appropriate sense, specifically on the transverse directions). These two principles reduce the freedom in the choice of the counterterms by constraining them to depend only on the induced metric g of BΣ, its extrinsic curvature k with respect to one of the null normals, and linearly on the variation of these On the other hand, g and k themselves depend only on the large gauge fields χ A , Φ, and C. This has an important consequence -it implies that we can never add a counterterm that depends on the mass and angular momentum aspects of the metric. They will necessarily enter the symplectic form and subsequently the charges. Two independent, but equivalent, procedures for the construction of this counterterm are described in Section 4.3.

Invertibility of the Symplectic Form and Resolution of Large u Divergences
The renormalization procedure described above cancels all the divergences at large r producing a symplectic potential that is finite as we take r Ñ 8. However, this still leaves behind large u divergences that we cannot remove with this new technique (this is discussed in Section 4.4). These large u divergences are of two types -there is an explicit divergence which is linear in u and an implicit divergence which arises from the dependence of the extrinsic curvature on u.
The explicit divergence is automatically removed when we fix the completely separate problem of invertibility. When computing the pre-symplectic form, we observe that it is degenerate. This can be understood intuitively by noticing (see equation (4.39)) that while the supertranslation mode Cpxq is conjugate to the mass aspect M pxq, and the superrotations mode χ A pxq is conjugate to angular momentum aspect N A pxq, other independent degrees of freedom that could potentially be conjugate to Φ are simply not available! We are thus forced to freeze this degree of freedom on the phase space. 4 We do this by freezing the volume form on I`. It turns out that this condition also gets rid of the explicit linear-in-u divergence in the symplectic potential. With this restriction, we construct the non-degenerate symplectic form in Section 5. We then invert this and construct all the Poisson/Dirac brackets of the theory explicitly (see equations (5.8)-(5.12)).
The implicit large u divergence cannot be renormalized and in fact plays a physical role in the charge algebra, discussed in detail in section 6.

GBMS Diffeomorphisms are NOT Canonical Transformations!
In Section 6, we attempt to construct the Hamiltonian charges which generate GBMS diffeomorphisms. Unsurprisingly (based on our four-dimensional intuition), we find that the charge is in fact, not integrable! More precisely, there is no Hamiltonian function H ξ associated to a GBMS vector field ξ on the phase space, that satisfies It follows that GBMS diffeomorphisms are not canonical transformations! The obstructions come precisely from the non-covariant pieces in the symplectic potential that are responsible for implicit large u divergences. It might be possible to solve this integrability issue by choosing field-dependent parameters for the GBMS diffeomorphisms, but this requires solving a complicated functional derivative equation, which is beyond the scope of this project. The usual way of handling this non-integrability is to work with local non-integrable charges. Here we take a different route.
Noting that we cannot construct charges associated to GBMS diffeomorphisms, we do the next best thing -we extract the "integrable part" of Ωpδ, δ ξ q and use that to construct the corresponding Hamiltonian charges. By construction, these functions would be the closest thing that we have to GBMS charges. This is done in Section 6 and the corresponding charges are where f and Y A are the field-independent supertranslation and superrotation parameters. Using the brackets (5.8)-(5.12), we find that they faithfully represent the GBMS algebra 10) and the transformations they generate are almost identical to the variations from GBMS diffeomorphisms (denoted in the equations below with δ ξ ) (1.11) The only difference is in their action on the angular momentum aspect. Under GBMS diffeomorphisms N A has a more complicated non-covariant transformation. We call these types of transformations GBMS transformations and distinguish them from GBMS diffeomorphisms.

GBMS Ward Identities and Soft Theorems
In Section 7.1, we quantize the classical phase space constructed in the previous section (using canonical quantization) and in doing so, we construct the asymptotic Hilbert space of the theory.
This has an infinite-dimensional moduli space of vacua which are spanned by the zero mode operators Cpxq, M pxq and χ A pxq, N A pxq. We next define creation and annihilation operators from the radiation field r h (2) (essentially by taking a Fourier transform in u Ñ ω) and construct the rest of the Hilbert space as a Fock space. From this point, we follow the standard procedure to construct the Ward identities. In Section 7.2, we impose a set of antipodal matching conditions between fields on I`and I´on the large gauge modes which immediately implies equality of I`and IǴ MBS charges. This equality can be massaged into a "soft theorem" by using Einstein's equations to recast the GBMS charges as integrals over all of I`which then allows us to separate out a soft part which is linear in the radiation field and a hard part which is quadratic. We then explicitly verify in Section 7.3 that the Ward identities constructed in this way are exactly the same as the soft-graviton theorems, including the non-trivial generalization to spinning fields.
2 Asymptotically Flat Spacetimes Near IÌ n this section, we define and classify the space of geometries that we study in this paper.

Definition
The Penrose diagram of a globally asymptotically flat spacetime is shown in Figure 2.1.
In this paper, we will be interested in studying the structure of asymptotically flat spacetimes at asymptotic future null infinity I`. 5 To describe these regions, it is convenient to work in Bondi-Sachs coordinates X µ " pu, r, x A q where the metric takes the form h AB satisfies the so-called Bondi determinant condition The metric functions U , β, W A and h AB are all functions of u, r and x A .
In asymptotically flat spacetimes, for a fixed coordinate value pu, x A q, the coordinate r extends to infinity and the conformal boundary at r Ñ 8 is I`. Near I`(i.e. at large r), the metric 5 Formally, I`is defined by first conformally compactifying pM, Gq to an unphysical spacetime pM, r Gq and then affixing I`as a finite null boundary of pM, r Gq. In this paper, we do not discuss this compactification and I`is simply taken to be an asymptotic null boundary of pM, Gq.
These conditions ensure that I`is null. The past and future boundaries of I`are located at u "´8 and u "`8 and denoted by I`and I`respectively.

Classification of Ricci-Flat Metrics
We now consider asymptotically flat geometries in Einstein gravity. These geometries are Ricci-flat, In this section, we will study the behaviour of these geometries near I`, i.e. near r Ñ 8. The complete analysis of the asymptotic structure in any dimension D ě 4 can be found in [30]. Here, 6 More general boundary conditions, which we will not work with, compatible with the requirement of a null boundary and Einstein equations are U " Oprq, β " Opr 0 q, W A " Opr´1q and hAB " qABpu, xq`Opr´1q. See [30] for the discussion of the solution space with such conditions.
we report the complete result for D " 6.

General (Polyhomogeneous) Solution Space: S general
The large r expansion for generic Ricci-flat metrics in D " 6 is given by Einstein's equations (2.4) determine the expansion and its coefficients as follows: • R rr rGs " 0 fixes β in terms of h tABu .
• R rA rGs " 0 fixes W A in terms of h tABu and W A (5) .
• G AB R AB rGs " 0 fixes U in terms of h tABu , W A (5) and U (3) .
• R AB rGs " 0 fixes h tABu in terms of h (2)tABu , W A (5) and U (3) and h tABu | Γ where Γ is the null surface defined by u "´8. Note that I`X Γ " I`.
• R ur rGs " 0 does not impose any new constraints.
Note that the first three sets of equations are algebraic in nature so they fix the corresponding functions entirely whereas the last three sets of equations are differential (in u) so they fix their corresponding functions up to u-independent integration constants. Here, we have chosen the values 7 We will use this curly bracket notation around spacetime indices to denote the symmetric trace-free part of a tensor X tABu " X pABq´1 4 qABtrrXs.
of the fields at a fixed null time u "´8 as the integration constants. 8 Altogether, the full solution is determined entirely by the following data: The complete details of the solution including explicit formulas for each of the large r coefficients shown in (2.5) can be found in Appendix B.1. The interpretation of each of the fields in (2.6) is as follows: • h (2)tABu describes the gravitational flux across I`.
• q AB is the transverse induced metric on I`.
• U (3) | I`d escribes the mass distribution in the system.
• W A (5) | I`d escribes the angular momentum distribution in the system.
To conclude, let us point out that (2.5) is constructed following a principle of "naturality". We have only included those log terms that are forced on us by the equations of motion. In this case, the independent log terms are h (2, 1)AB , U (3,1) and W A (5, 1) . Every other log term in the asymptotic expansion is sourced by these three modes. These three "fundamental" log terms are themselves sourced by the non-log terms in the expansion, e.g. h (2, 1) is sourced by h (1) so we cannot set these to zero without imposing additional constraints on the solution. This procedure naturally produces maximally polyhomogeneous expansions starting from the radiation order (for example in the case of four-dimensional spacetimes it shows the generic existence of a time-independent term h (1, 1) ).
Notice that this approach is different from that pursued in [46,47], where the solution space is defined starting from an arbitrary choice of polyhomogeneous initial characteristic data.

Analytic Solution Space: S analytic
Due to several large r divergences, it is very difficult (though perhaps not impossible) to elevate the polyhomogeneous solution space of the previous section to a phase space. The main culprit turns out to be the log terms in (2.5). As we will see in Section 4, in the absence of such terms it is possible to construct the phase space and even though the aforementioned large r divergences are still present, it is possible to regulate and renormalize them.
The log terms can be removed by imposing the following diffeomorphism invariant constraint r 4 e μ µ e ν ν e ρ ρ e σ σ R µνρσ rGs| I`" 0.
Here e μ µ is an orthonormal basis of vielbein. Physically, we interpret this condition to imply the absence of gravitational flux through Γ near I`. In Appendix B.2, we present a detailed analysis of this constraint and show that it does, in fact, remove all the log terms. In addition to this, it also imposes a few constraints on the non-log coefficients which we now discuss.
The first non-trivial constraint is the vanishing of the Weyl tensor of q AB which implies that it is conformally flat, Φpxq is the Weyl factor and χ A pxq is the finite diffeomorphism which maps the flat metricq AB to the flat Cartesian metric δ AB . From the bulk point of view, χ A is a large diffeomorphism (as we will see in Section 3.1) so it is not trivial! Next, (2.7) imposes a constraint on h (1) , where D A is a covariant derivative w.r.t. q AB . Here and in the rest of this paper, we employ vector and matrix notation where all repeated indices are contracted w.r.t. q AB , e.g.
Derivative indices are similarly contracted Together with conformal flatness of q and the equation of motion R AB rGs " 0, (2.9) solves to In other words, h (1) is determined entirely in terms of a scalar Cpxq. 9 It is important to note that q AB and h (1)AB are the actual relevant degrees of freedom in the geometry, so Φpxq, χ A pxq and Cpxq are not independently uniquely defined. Rather, the triò χ A pxq, Φpxq, Cpxq˘is defined up to the identification where p2D tA D Bu`RtABu qre Φpxq cpxqs " 0. (2.14) In other words, x g is a conformal transformation and Ω g is the corresponding Weyl factor. It is clear that such a redefinition leaves q and h (1) invariant. 9 h (1) is defined in terms of e Φ C instead of C as this simplifies many of the equations derived in this paper.
To describe the final set of constraints imposed by (2.7), it is useful to first define the following where the superscript TF denotes the trace-free part w.r.t. q. The constraints can now be written With this, we are done! Once all the constraints described above are imposed, all the log terms vanish and we have an analytic solution.
Before ending this section, we note that while (2.7) forces the tilded fields (2.15) to vanish on I`, it does not tell us how fast those fields fall-off in a neighbourhood. However, it is possible to determine this using other considerations which are outlined in Section 2.4. For completeness and readability, we retroactively present those results in this section. First, it is noted that the tilded fields (2.15) satisfy Using this, we can determine the large u behaviour of U (3) and W (5) as well. To do this, we first They give an appropriate definition of the mass and angular momentum aspect. Using (2.17), we can immediately determine the large u behaviour of these fields The behaviour at large positive u is fixed by the requirement that the system reverts to the vacuum in the far future of I`. 11 M pxq and N A pxq are the integration constants that are obtained upon integrating the u-evolution equations.
With all of this, we can now summarise the data that defines the analytic solution space . (2.20) The detailed asymptotic expansion for solutions in S analytic is given in Appendix B.2.

Canonical Solution Space: S canonical
The two solution spaces described so far are novel and apart from [30], have not been previously discussed in the literature. The standard or "canonical" solution space, which has been extensively studied, e.g. in [18,20,34,48,49], involves a much stronger definition for asymptotically flat spacetimes (compared to the one we are employing here) which is motivated by the asymptotic behaviour of linearised gravitational radiation, namely 12 This implies that q AB " δ AB and h (1)AB " 0. Comparing this to (2.8) and (2.12), these boundary conditions are equivalent to the requirement Cpxq " Φpxq " 0 and χ A pxq " x A . The data that defines this solution space is therefore Note that we have used a superscript˝to distinguish metrics in S canonical from those in S analytic .
The detailed asymptotic expansion for solutions in S canonical is given in Appendix B.3.

Linearised Gravity and Mode Expansions
Having discussed the general (non-linear) solution space in Einstein gravity, we now discuss solutions in linearised gravity in the background of Minkowski spacetime. These solutions are described by a mode expansion and live in S canonical . To construct the solutions, we first write the metric as This condition is modified if there exist massive matter fields in the theory which contribute a non-vanishing flux at I`. In this case, both r U (3) and Ă W (5) are non-vanishing at I`. 12 This is not entirely correct. The definition that is actually used is U " 1`Opr´2q, β " Opr´4q, W A " Opr´3q and hAB " γAB`Opr´2q where γAB is the round metric on S 4 . Here, we use a slightly different but equivalent definition. This choice simplifies our calculations since we do not have to keep track of curvature contributions from the round S 4 . η µν pXq is the metric of Minkowski spacetime in Bondi-Sachs coordinates. The linearised perturbationγ satisfies the equation This equation is traditionally solved in harmonic gauge (H) where we have This is the standard wave equation. In Cartesian coordinatesX µ , the radiative solution to the wave equation is given by a mode expansion where p 2 " p µ ε AB µν ppq "η µν ε AB µν ppq " 0. The solution in (2.26) is given in Cartesian coordinates (for the backgroundη) and in harmonic gauge (for the perturbationγ). In this paper, however, we are interested working entirely in Bondi-Sachs gauge. To do this, we simply need to perform the appropriate coordinate and gauge transformations. The full solution in Bondi-Sachs gauge takes the form Here,X µ pXq is the coordinate transformation that maps from Cartesian coordinates to Bondi-Sachs coordinates. This is given bȳ The second term is required to map from harmonic gauge to Bondi-Sachs gauge. The vector field ζ µ which does this will be determined by imposing the Bondi-Sachs gauge conditions, (2.29) The integration constants in r are fixed by requiring that ζ µ is a small diffeomorphism so that To solve (2.29), we first need to determineγ (H) . We parameterize the integration variable p and polarization tensor εppq as With this choice, it can immediately be determined thatγ (H) uµ " 0 and ȷ . (2.32) With this, we can solve the differential equations (2.29). To do this, we first perform the integral over y above. This is given by To evaluate this, we first do a Taylor expansion of O AB pω, x`yq around y " 0 and then use the We then find (2.35) Using this large r expansion, we can now determine ζ α from (2.29) and (2.30), (2.36) Using this, we can determine the large r expansion forh AB " 1 r 2γAB in Bondi-Sachs gauge, and a 1 and a 2 are solutions to the quadratic polynomial With this, we can finally extract the mode expansion formulae forh (2) ,h (3) andh (4) as ı . (2.40) Note that the mode expansion forh (2) is valid in any gauge but the formulas for the subleading objects depends on our gauge choice. We also note that the equations above are consistent with the linearised versions of (B.51).

Large u Fall-Offs
In this subsection, we provide two arguments for determining large u falloff conditions of each coefficient of h AB . The first is based on the linearised analysis performed above the second is on the consistency of the solution space under BMS diffeomorphisms.

From Linearised Gravity
Having determinedh AB as a mode expansion in (2.40), we can use it to determine the large u fall-offs of these fields. It is clear that at large u, the dominant contribution to these integrals arises from the region near ω " 0. To perform the integral, we expand the operator O AB as This structure of the expansion around ω " 0 follows from our understanding of the soft (low energy) behaviour of graviton scattering amplitudes. In particular, insertions of the first two operators shown above are related to the leading and subleading soft graviton theorems respectively which are known to be universal, i.e. they do not depend on any details of the theory [5,10]. We will review these soft theorems in Section 7. It is also possible to see that the operators satisfy [28] where D paq are defined in (2.38). These differential operators are precisely the ones that show up in the mode expansions (2.40) and (2.42) play a crucial role in ensuring that the integrals over ω converge.
Substituting (2.41) into the integrands of (2.40), we find (2.43) It follows thath With this, we have determined the large u fall-offs for linearised solutions in S canonical . However, since the fields die off at large u, it immediately follows that (2.44) hold in the non-linear theory as well (at least to all orders in perturbation theory). Finally, using the fact that solutions in S analytic are obtained from those in S canonical via a WBMS diffeomorphism (which we will prove later in Section 3.2) and that under this maph Ñ r h, we can conclude that (2.44) generalises as-is to S analytic so we have

From WBMS Consistency
Another argument in support of (2.45) comes from the reasonable demand that the solution space S analytic is consistent with WBMS diffeomorphisms. 13 More precisely, we have to ensure that the boundary condition (2.7) is not violated by WBMS diffeomorphisms. To run the argument, we need to consider the component

46)
13 If the reader is not familiar with WBMS diffeomorphisms, they should skip this section and study Section 3 first.
of the condition (2.7). The reason for this is that the leading order terms of this component of the Riemann tensor are functions of r h p2q , r h p3q and r h p4q . Explicitly With a slight anticipation with respect to Section 3, infinitesimal WBMS diffeomorphisms are generated by the vector fields where crucially the function T is linear in the coordinate u. The action of this vector field on the Riemann tensor is given by (2.49) Notice that because of the linear u dependence of both T and h p1qAB , we need the following to be true in order for the WBMS diffeomorphisms to preserve the boundary conditions These fall-offs are consistent with (2.45). Note that the fall-offs derived here are weaker than (2.45), but in fact all we will need in the subsequent calculations of this paper.

Summary and Discussion
Before moving on we take pause, summarise the results of this section, and contextualize them within a historical framework.
We started in Section 2.2.1 by noting that the most general asymptotic expansion admissible by Einstein's equations is polyhomogeneous near I`. In the context of the asymptotic symmetry analysis, some forms of polyhomogeneous solutions were first discussed in four-dimensional gravity in [50][51][52][53] and the log terms are now understood to be related to infrared divergences which plague all four-dimensional theories with massless particles. Such infrared issues are absent in higher dimensions so in this case, we must find a new interpretation of the log terms.
To do this, it is useful to take inspiration from AdS/CFT where log terms in the asymptotic expansion of the bulk solution are associated with anomalies in the dual CFT. For example, the asymptotic expansion in Fefferman-Graham gauge for generic asymptotically AdS solutions in D " 5 has the form where r is the radial Poincaré coordinate. The presence of the log term presents an obstruction to holographic renormalization and one is forced to introduce non-covariant terms in the counterterm action [40]. This non-covariance then generates a Weyl anomaly in the dual CFT. A similar argument will likely apply to the log terms that appear in (2.5). In the present case, there are three fundamental log terms: h (2, 1)AB , U (3, 1) and W (5, 1)A and these are likely related to the conformal anomaly, supertranslation anomaly and superrotation anomaly respectively. We will explore these interesting issues in future work. 14 We next constructed analytic solutions in Section 2.2.2 by imposing a "no-flux condition" (2.7) thereby showing that the no-flux condition is related to the log terms. We can compare this with some well-known facts and theorems developed in four-dimensional gravity. In four-dimensional gravity, a sufficient condition for smooth initial hyperboloidal data 15 (which then develops to a nonpolyhomogeneous I according to [54]) is that the conformal Weyl tensor at the boundary of the hyperboloidal initial data surface vanishes [55]. It is also known that non-trivial solutions of Einstein equations exist that satisfy the peeling property [56] thanks to the Corvino-Schoen theorem [57].
Less formally, it is known that early-time waves are a source for the polyhomogeneity of I` [58].
Although the setting in which we work is not exactly the same as the aforementioned works (for example we only gave the falloff rate of the radiation field toward I`, rather than specifying a stationary portion of I), it is likely that S analytic can be obtained by adapting those constructions to higher dimensions. We also note that the analytic solution space S analytic is actually compatible with (a higher dimensional generalisation of) the peeling theorem [19][20][21].
Finally, in Section 2.2.3, we discussed the canonical solution space S canonical . Historically, this is the solution space that was most widely studied in higher dimensional gravity essentially due to the fact that it is the simplest and most natural generalization of the linearised solutions which we reviewed in Section 2.3. This consideration motivated previous works to argue in favour of setting h p1qAB " 0 on all of I`and it is this restriction which led to the conclusion that the asymptotic symmetry group for higher dimensional gravity is the finite-dimensional Poincaré group.
Another argument in favour of setting h (1) " 0 is the following. Consider the solution space S canonical augmented just by the inclusion of h (1) . Here, h p1qAB is time independent and pure gauge (compare with (2.12)). If the system undergoes a stationary-to-stationary transition, then the same supertranslation can be used to set h p1qAB " 0 on all I [34]. This argument assumes that the supertranslation involved in the process is a small gauge transformation. However, a straightforward evaluation of the supertranslation charge in such a phase space gives a divergent result. Without a proper renormalization of the charge, there is no reason to draw the above conclusion. Indeed, the phase space renormalization procedure we develop in this paper suggests that in S analytic the gauge fixing of h (1) corresponds to a large gauge choice.
To summarise, we have spelt out the conditions and constraints near spacelike infinity that the gravitational field has to satisfy in order to evolve to the form given in (B.35). We thus have a constructive definition of the solution space at null infinity which serves several purposes: piq clarify and remove unnecessary assumptions from previous literature, piiq give a detailed understanding of the action of asymptotic symmetries on this solution space, piiiq provide the basis for analysing what conditions at spacelike infinity may give rise to the given structure at null infinity, pivq assess holographic structures, such as anomalies in the dual theory as suggested in [30] and pvq give the building blocks for further consistent extensions of the phase space.

Weyl-BMS Diffeomorphisms
In this section, we determine the asymptotic symmetry group for the space of solutions described in the previous section. This is the set of all diffeomorphisms that preserve the gauge conditions (2.1), (2.2) and boundary conditions (2.3) and which act non-trivially on the data of the theory.

The Weyl-BMS Algebra
Infinitesimal diffeomorphisms are generated by vector field ξ µ and act on the metric via the Lie To preserve the gauge conditions, we must have whereas in order to preserve the boundary conditions, we must have To solution to these equations is relatively easy to find and in general are parameterized by functions f pxq, ωpxq and Y A pxq, where The vector field (3.4) satisfies the algebra r , s ‹ is a modified Lie bracket which keeps track of the fact that the BMS vector field depends on the metric. Equation (3.8) In the above, L ξ is the 4D Lie derivative w.r.t. the vector field ξ A . Expanding this in large r, we can determine the action of the WBMS vector field on the data of the theory. For instance, on the leading large r coefficients of h AB , we find Note that Y A generates an infinitesimal diffeomorphism on the metric q AB and ω generates an infinitesimal Weyl transformation.
On the analytic solution space S analytic , these diffeomorphisms act as (3.10)

Important Subalgebras
The WBMS algebra has three important subalgebras: (1) Poincaré Algebra: This is generated by vector fields which preserve q AB and which act covariantly on the u-independent part of h (1) . These satisfy This generates a finite-dimensional subalgebra which is a generalisation of the Poincaré algebra. To reproduce the Poincaré algebra specifically, we consider the case where q AB is a conformally flat metric. Then, moving to coordinates where the metric has the form q AB " e 2Φ δ AB , we find the solution Y A pxq is a conformal Killing vector and generates the Lorentz subgroup SOp1, 5q. f pxq generates the translation subgroup T 6 . α generates time translations and pβ A , γq generates the spatial translations.
The algebra of these generators is precisely T 6¸S Op1, 5q.
In older literature dealing with higher dimensions [34], the Poincarè algebra was not obtained as a subalgebra of a larger algebra, but rather as the full algebra of asymptotic symmetries. This arose from constraining oneself to the canonical solution space S canonical where h (1) " 0. This then automatically forces the equations (3.11) implying that the full asymptotic symmetry group is simply the Poincaré group.
(2) BMS Algebra: This is generated by vector fields which preserve q AB . They satisfy the first two equations in (3.11) but not the last. Consequently, for this subalgebra, Y A is still a conformal Killing vector w.r.t. q AB (which generates a finite-dimensional group) but f is an arbitrary function. The diffeomorphisms generated by f are known as BMS supertranslations (ST). The corresponding algebra is an infinite-dimensional generalisation of the Poincaré algebra and has the structure ST 6¸S Op1, 5q.
Since the celestial manifold is of dimension greater than two, all conformal Killing vectors are globally defined and hence there is no immediate generalisation of the algebraic notion of Virasorosuperrotations. A way to generalise this notion to higher dimensions has been explored in [29], by mapping codimension-2 cosmic branes to Bondi-Sachs gauge. In our six-dimensional setting, the explicit map can be easily read from Appendix C by specifying the cosmic brane tension parameter (which is the only parameter that differentiates this spacetime from pure Minkowski). These are spacetimes where the regularity of the celestial sphere is broken and a conical deficit is introduced on two-dimensional sections of the celestial manifold. They are intended to generalise to higher dimensions the relationship between cosmic strings and Virasoro-superrotations [59][60][61]. However, in four dimensions this relationship is supported by Penrose cut-and-paste constructions of impulsive waves. The Virasoro-superrotation is the map relating the spacetime on the two sides of the impulse.
For example, the dynamical process creating the shock is the breaking of a cosmic string. In higher dimensions such constructions are still missing.
(3) Generalised BMS Algebra: This is generated by vector fields which preserve the volume form on q AB , i.e. δ ξ ? q " 0. This satisfies only the first equation of (3.11) which fixes ω in terms of Y A but there is no additional restriction on Y A (so it is an arbitrary vector field). The corresponding diffeomorphisms generated by Y A pxq are (also) known as superrotations (SR). This algebra has the structure ST 6¸S R 4 .

Finite WBMS Diffeomorphisms from S canonical Ñ S analytic
Due to the particularly simple nature of (3.10), we can determine the structure of the finite WBMS transformations: The infinitesimal transformations are recovered by setting X A pxq " x A`Y A , Fpxq " f pxq and Wpxq " ωpxq and then expanding to linear order in Y , f and ω.
An important corollary of (3.13) is that it is always possible to perform a WBMS transformation to map a solution in S canonical (which has C " Φ " 0, χ A " x A ) to a generic solution in S analytic by choosing X " χ, F " C and W " Φ. These WBMS diffeomorphisms can be extended to the bulk so that for any metric G P S analytic , there exists a unique metricG P S canonical and a diffeomorphism X µ ÑX µ pXq such that G µν pXq " B µX ρ pXqB νX σ pXqG ρσ pXpXqq. (3.14) We performed a detailed analysis of these equations in Appendix C. In this section, we only present a couple of results that will be used in the next section.
With this goal in mind, firstly we clarify the notation that we are forced to employ in this section (and also in Section 4.1 and Appendix C). The confusing aspect of these sections is that we are simultaneously discussing solutions in S analytic and in S canonical for which the conformal metric on I`is q AB " e 2Φq AB and δ AB respectively. With three different concurrent metrics -namely q AB , q AB and δ AB -we have to be extra careful! We start by denoting any tensor in the solution space S canonical with a superscript˝, e.g.
The hatted tensors indices are raised and lowered w.r.t. the flat metricq AB . Note that by their definition, the hatted tensors have the same large u behaviour as the ringed tensors.
Finally, the unhatted tensors are related to the hatted ones by an extra Weyl factor where ∆ T is the scaling dimension of the tensor T . 16 These unhatted tensor indices are raised and lowered w.r.t. q AB .
With these definitions, it is easy to derive the following properties D uTA 1¨¨¨An pu, xq " B A 1 χ B 1 pxq¨¨¨B An χ Bn pxqB κTB 1¨¨¨Bn pκpu, xq, χpxqq, where we have defined the covariant derivatives by Here,D A is the covariant derivative w.r.t.q AB . Note that the covariant derivatives all commute and satisfy e´ΦD u p¨q " B u p¨q, e´ΦD A p¨q "´B u rB A κp¨qs`D A re´Φp¨qs. (3.21) With this notation, we can easily transform any equation for the ringed fields to an equation for the hatted fields with the following simple replacements  We now present our results. The finite WBMS transformationsXpXq satisfying (3.14) admits a Taylor expansion in r´1 of the form u " κ`e´Φů (1) r`e´2 (3.24) In order to find the relationship between the un-hatted tensors which appear in the asymptotic expansion of the metric, we need to determine all the coefficients shown above. This is done in Appendix C. For the purposes of section 4.1, we only need the very leading order results and

Gravitational Symplectic Potential
In this section, we will use the covariant phase space formalism [62][63][64][65][66][67][68] to elevate the solution space S 0 to a phase space Γ 0 . The dynamics of the theory is described by the Einstein-Hilbert Lagrangian The boundary term is typically required to make the variational principle well-defined, e.g. the Gibbons-Hawking-York boundary term. Varying the action, we find where r‹θpδqs µ " G µν δΓ ρ νρ rGs´G νρ δΓ µ νρ rGs. 3) The first term in (4.2) gives us the equations of motion (2.4). The (pre-)symplectic potential on a Cauchy slice Σ is obtained by integrating the boundary term in (4.2) over Σ, We see that there are two types of ambiguities in the pre-symplectic potential.
The first ambiguity is the δ-exact term which is related to the boundary term in the action (4.1).
Part of L B is fixed by the variational principle (e.g. Gibbons-Hawking-York boundary term). The rest of L B (typically referred to as the counterterm action) is fixed by requiring that the on-shell action be finite. In asymptotically AdS spacetimes, there is a systematic procedure to fix L B known as holographic renormalization. However, such an algorithm is lacking for flat spacetimes and we will not address this issue in this paper either. Here, we are interested in constructing the classical phase space of general relativity whose local structure depends crucially on the symplectic form Ω, not the symplectic potential. Any gauge-invariant δ-exact term in Θ never contributes to Ω.
Consequently, we will simply drop any such term that shows up in our calculation of the symplectic potential, including L B .
The second ambiguity is denoted Y in (4.4). Note that this is a boundary term and it arises from the fact that the variation of the action depends directly only on dθ. Consequently, a shift of the type θ Ñ θ`dY does not change (4.2). This ambiguity will play a crucial role in our renormalization procedure.
To understand this in more detail, let us use (3.14) to simplify θpδq. We first note that given (3.14), the variation of the metric has the form Note that since both the LHS and the first term above satisfy the Bondi-Sachs gauge conditions and our boundary conditions for asymptotic flatness, the second term must satisfy the same conditions. However, these are precisely the conditions (3.2), (3.3) that define a WBMS vector field.
Consequently, { δX has the form (3.4) and is parameterized by functions f , ω and Y A . They can be determined from the leading order behaviour of the finite WBMS transformations Using (4.5), we can rewrite the θpδq as where r‹θ can pδqs µ " G µν B νX ρ´r ‹θpδqs ρ | XÑX¯, r‹Q ξ s µν " 2∇ rµ ξ νs .
Here,θ is evaluated in the canonical solution space S canonical detailed in Section B.3. Since this solution space only allows for gravitational radiation, θ can represents the radiative contribution.
Using (4.7), we can rewrite (4.4) as As we will see, the first term is completely finite on I`. All the divergences appear in the second term above and these will be cancelled by a judicious choice of Y pδq.

Bulk Contribution (I`)
We wish to evaluate the symplectic potential on the asymptotic null boundary I`whereas the formula (4.9) can only be applied to finite null boundaries. To fix this, we define a timelike regulated boundary Ir by r " constant. We then define the symplectic potential on I`by now we have to replace X ÑX everywhere and rewrite everything in terms of the hatted fields defined in (3.15). This can be done quite simply using the replacement rules (3.22) though in addition to those, we also need a replacement rule for the variation δ. This is easily done using the definition (3.15) from which we find (4.14) It follows from (4.13) that the replacement rule for δ is simply δ Ñ { δ. As before, { δ commutes with the derivatives D u and D A and with the flat metricq AB so the order of the replacement is irrelevant.
We also note that it satisfies the property Putting all of this together, we find r‹θpδqs r | XÑX¯" Opr´6q, where we have usedh (2) "ĥ (2) as shown in (3.25). Let us process each of these terms one-by-one.
To do this, we will decompose the variation { δ into two pieces as The reason for this decomposition will become apparent shortly. Using this, the first term becomes Next, using (4.15), we can simplify the second term as where we have used (3.26). The first term is δ-exact and will not contribute to the symplectic form so we shall henceforth drop it. The last two terms are linearly divergent at large u. To deal with this, we have regulated the boundary I`by setting it to u " u 0 instead of at u "´8. We are eventually interested in taking u 0 Ñ´8. Such large u divergences will be discussed in Section 4.4.
The third term in (4.17) is a total u-derivative which then vanishes due to equations of motion (B.51) and the large u fall-offs (2.45). The last term can be manipulated as (4.21) The last term vanishes due to the constraint (3.23). The first term simplifies to a total derivative due to (3.21) as can pδq " (4.23)

Boundary Contribution (I`)
Having discussed the canonical contribution in the previous section, we now turn to the second term in (4.9) which we recall here The boundary BIr is located at u " u 0 so the first term above simplifies to For a WBMS vector field ξ (3.4), we have To recast (4.26) into the form (4.27), we have used the equations listed in Appendix B.2 quite extensively. Note that the term on the RHS is evaluated at I`(regulated at u " u 0 ) so we can also use the boundary conditions (2.17) to simplify our results. In particular, this implies that only terms that depend on q AB and h (1)AB appear in the boundary symplectic form as can be seen in (4.27).

Large r Divergences and Phase Space Renormalization
Unlike the radiative/bulk contribution we discussed in the previous section, the boundary contribution to the symplectic potential is divergent at large r. In this section, we show that via a judicious choice of Y pδq, the large r divergences (and only the divergences) can be cancelled. Such a renormalization process was discussed in four dimensions in [43,44], where it was suggested that since the divergences in the less involved four-dimensional analogue of (4.27) are all boundary terms, we can simply cancel them all by choosing Y pδq "´divergences. However, there are two major issues with the prescription (4.29) which we will address in this section. 17 Due to the constraints (2.16), the fields and variations satisfy • It is crucial that Y pδq is covariant which means that it must be constructed out of the induced metric on BΣ and the extrinsic curvature (which keeps track of the embedding of BΣ ãÑ M ).
It implies that the symplectic potential will not depend on our choice of regulator for the asymptotic surface BΣ. For example, in this paper, we choose to regulate I`by a r " constant surface. Of course, this is only one out of many choices of regulators. For instance, we can define the regulated surface in the same way but in Newman-Unti coordinates [69] or as a v " constant surface where v is a null coordinate, etc. As long as Y pδq is constructed in a covariant way, the final symplectic potential does not depend on this choice. Secondly, in order to preserve bulk locality, it is also crucial that Y pδq be constructed locally out of the above-mentioned quantities. This is the first major problem with (4.29). It is not at all obvious that there even exists a local and covariant choice of Y pδq that cancels all the divergences.
• A second major issue with (4.29) is regarding the finite terms in (4.27). If we were indeed allowed to cancel all the divergences through such a simple prescription, what is stopping us from using the same prescription to cancel the finite terms as well? We would then end up in a phase space with no BMS transformations! It would then seem that the existence of BMS symmetries in the gravitational phase space is entirely a matter of choice.
In this section, we will address both the issues above and show that by enforcing the locality and covariance of Y pδq and by using the boundary conditions (2.7) which defines the solution space S 0 we are indeed able to cancel all the divergences, but not the finite part!

General Structure
We start by making the implications of locality and covariance on Y pδq more precise. First, locality of Y pδq implies that it has the structure where Ypδq is a scalar function of its arguments and ds 2 | BΣ " g AB dθ A dθ B is the induced metric on BΣ. Further, covariance implies that Ypδq can only depend on the induced metric, the extrinsic curvatures and their variations. In particular, a Euclidean codimension-2 surface admits two independent null normals n and ℓ (with n¨ℓ "´1) and we can define two extrinsic curvatures as where e µ A is the projection tensor from M Ñ BΣ. On-shell, one of the extrinsic curvatures is related to the other. In particular, if ℓ is tangent to the null surface whose boundary is BΣ, then l is fixed in terms of k and g. Consequently, we can simply work with the latter two quantities. Indeed, in the case at hand l AB " B u g AB which is fixed by the equations R AB rGs " 0. On the other hand k AB " B r g AB is not fixed by any equation.
Locality and covariance together imply that Ypδq is a polynomial function of the following arguments Ypδq " Y AB 1 pg, D n kqδg AB`Y AB 2 pg, D n kqδk AB . (4.32) where D is the covariant derivative w.r.t. the induced metric g AB . The goal of this section will be to construct a Y of this form so that all the divergences are cancelled from (4.27). Let us now apply the discussion of the previous section to the case at hand. The induced metric on BIr is g AB " r 2 h AB . The two null normals are given by We must now construct Ypδq which cancels the divergences in (4.27). We first note that at large r, the metric and extrinsic curvature have the form and we define the covariant tensor 18 In this section, to preserve covariance, we raise and lower indices using the induced metric g.
From (4.34) and (4.35), we see that the non-covariant tensors q AB and h (1)AB are the leading order large r coefficients of the covariant tensors g AB and h AB . This strongly suggests that any instances of q AB and h (1)AB in a divergent quantity can be covariantized by replacing Of course, it is important to note that q and h (1) are only the leading order coefficients of g and h so making such a replacement modifies the subleading divergent terms in the large r expansion. Consequently, our analysis must be done order-by-order in large r. Following this process, it can be shown that the appropriate counterterm is given by (4.37) 18 To make contact with the notation in [30], notice that kAB here is rkAB there and that hAB here is 2rKAB there in equations (4.1)-(4.2). Einstein equations can be written in terms of such tensors and lAB " Bu gAB{2, see equations (4.6)-(4.10) there.
The counterterm above cancels all the terms except for the last four in (4.27). Above, we have also included a finite counterterm Y finite which changes the symplectic potential by a finite amount.

The most general form for such a counterterm is
? gY finite pδq " ? gˆa 1 trrh 2 sph 2 q tABu`a 2 trrh 3 sh AB˙δ g AB δ ? gˆa 3 trrh 4 s`a 4 trrh 2 s 2˙`δˆa 5 ? gtrrh 4 s`a 6 ? gtrrh 2 s 2˙. (4.38) Here, we have classified the ambiguities into three types of terms. The last two are δ-exact terms so they do not contribute to the symplectic form and we will henceforth drop them. The middle two terms also do not contribute to the symplectic potential for reasons we will discuss in the next section. Finally, the first two terms are the non-trivial ones which modify the BMS charges and their algebra.
Putting together the radiative and boundary contributions, we find that the renormalized symplectic potential (including all the finite ambiguities) has the form where N A pxq " N A pxq´P A pxq and P " 1 24ˆh where h (1) above is evaluated at u " u 0 .
Alternative way For a quicker construction of the counterterm, we can take advantage of a simple computational trick. Knowing that the divergences are a boundary term it is easy to show that [43,44] ż u"const p‹θq u div dr " ż r"const p‹θq r div du. (4.41) In this paper we work with metrics that have an analytic expansion in terms of r. This implies that the entire ş p‹θq u dr is a boundary term. Therefore, ş p‹θq u dr has the same tensor structure as the counterterm we need to add to Θ to make it finite.
The question now is how to express ş p‹θq u dr covariantly in terms of g AB and k AB and their variations. This is achieved in two simple steps. The first is the following, (4.42) Here we note two things: firstly, the awkward δ ? g will be fixed to 0 in the subsequent analysis.
Even if it was not, it could be fixed by adding an additional term L 3 n and re-adjusting the coefficients of the L n , n P t0, . . . , 3u. These would make the calculations in this section more tedious and since it is ultimately unnecessary it is not done here. Secondly, there will be no finite order contributions from this counterterm. The reason for this is that in the first line above r´1 integrates to ln r and the coefficient of this term necessarily has to vanish due to the no-log constraint.
The second step is expressing r derivatives of g AB and k AB in terms of themselves. This is easily done by using the equations of motion, boundary conditions (specifically all we need is the fact that r 4 R rArB rGs " 0| I`) and the definition of k AB (4.43) Notice that even though the third equation does not hold in general, it is valid to the orders relevant to this computation. This is not an accident, and one can prove that a relationship of this type holds in any dimension.

Large u Divergences
In the previous section, we have discussed the large r divergences and their renormalization extensively. We now turn to the large u divergences in the symplectic potential.
There are two types of u dependences in Θ I`p δq. First, an explicit factor of u 0 appears in the second line of (4.39) and second, there is an implicit u 0 dependence in h (1) . The implicit dependence cannot be renormalized and in fact, plays a physical role in the charge algebra. The divergence arising from the explicit u dependence can be removed by restricting to a smaller phase space where While the reasoning provided here is sufficient to impose such a constraint, there is a second reason for expecting this. From (4.39) we can easily extract the conjugate modes for all the fields on our phase space. In particular, the symplectic conjugate of C is M and that of χ A is N A (roughly).
However, the Weyl mode Φ (which appears in δ ? q) does not have any independent symplectic conjugate so the symplectic form constructed out of (4.39) will be degenerate. In order, then, to have a well-defined phase space, we must remove Φ as a dynamical variable and consequently fix its variation in terms of other variations, i.e. we must require The precise choice of the function F is required to preserve the identification (2.13) and (4.46) does the job! In this case, we can simply solve for Φpxq as Φ 0 pxq is a non-dynamical background Weyl factor that is constant over the entire phase space. On such a phase space, the symplectic potential is

Symplectic Form and Poisson Brackets
The symplectic form is constructed from the symplectic potential as Ωpδ, δ 1 q " δΘ I`p δ 1 q´δ 1 Θ I`p δq´Θ I`p rδ, δ 1 sq.
Note that Ω does not depend on the choice of Cauchy slice, even though Θ does so we have dropped the subscript I`from Ω. Using this definition, the symplectic form can easily be constructed from (4.49). Our goal is to eventually invert the symplectic form in order to construct the Poisson brackets on the phase space. In order to do this, it will be useful to work in a particularly nice basis where the inversion becomes trivial. To do this, we define a primed field h 1 (2) by Using this, the symplectic potential can be written as (5.5) Using (5.3), we find that the symplectic form is given by where N is the wedge product in field space, δf N δg " δf δ 1 g´δ 1 f δg. From (5.6), we can see the usefulness of defining the primed fields. In terms of these, the symplectic form attains a simple block diagonal form! Inverting this becomes incredibly simple and the Poisson brackets are given by th 1 (2)AB pu, xq, B u 1 h 1 (2)CD pu 1 , x 1 qu " 16πGδ AtC δ DuB δpu´u 1 qδ 4 px, x 1 q, all others " 0. Note that these brackets are consistent with the identification (2.13).
We can now go back to the original unprimed fields and determine the full set of Poisson brackets.

GBMS Charge Algebra
Now that we have constructed the symplectic form for six-dimensional gravity, we turn to a study of canonical transformations on this phase space. We recall that a transformation δ c which acts on the fields is canonical iff. there exists a Hamiltonian charge which satisfies δH c " Ωpδ, δ c q. (6.1) Particularly important transformations that we would like to consider are the WBMS diffeomorphisms discussed in Section 3. Under infinitesimal WBMS diffeomorphisms, the phase space fields transform as in (3.10) which we recall here Using (5.6), we find To process this equation, we need to evaluate the GBMS diffeomorphism of the primed fields. We start by recalling all the relevant definitions here (from (5.2) and (5.5)) 6) where N A is related to N A via (4.40) Using these definitions, we easily find (6.8) Substituting these into (6.5), we find The LHS is δ-exact so the Hamiltonian charge H ξ exists iff. the RHS is δ-exact. We see that the first two terms above are δ-exact if we assume that the GBMS parameters f and Y A are constants on the phase space so that δf " δY A " 0. On the other hand, the last term presents a serious obstruction to exactness. It follows from this that GBMS diffeomorphisms are in fact, not canonical transformations on the phase space and the charge H ξ does not exist! 19 Despite the fact that GBMS diffeomorphisms are not canonical, we can still proceed to construct some charges which have a GBMS-like structure. Inspired by the form of equation (6.9), we define the following charges Using the commutators (5.8)-(5.12), we can immediately check that these charges satisfy the BMS algebra (3.6) 19 A potential way out of this dilemma is to not take the parameters f and Y A to be constant on the phase space but to choose their dependence so as to cancel the last term in (6.9). We do not consider this in this work.
The action of these charges on the phase space variables are We shall refer to the transformations that these charges generate as "GBMS transformations". These must be distinguished from the GBMS diffeomorphisms which act as (6.2) with (6.3). GBMS transformations differ from GBMS diffeomorphisms by their action on the N A ! Indeed, the "difference" between the transformation generated by the charges and GBMS diffeomorphisms is As expected, the "extra" term in the GBMS transformation of N A (as compared to GBMS diffeomorphisms) is precisely the obstruction term in (6.9).

GBMS Ward Identities and Soft Theorems
In this section, we construct the semi-classical Hilbert space by quantizing the classical phase space constructed in Section 5. This involves elevating all functions on the phase space to operators, Poisson brackets to quantum commutators, t , u Ñ 1 i r , s and complex conjugation to the adjoint. The Hilbert space furnishes an irreducible representation of the Poisson bracket algebra (5.8)-(5.12).
The procedure implemented here is similar in structure to the one described in Section 4 of [68].

Quantum Hilbert Space
We start by discussing the vacuum sector. This is spanned by the operators Cpxq, M pxq, χ A pxq and N A pxq. The operators C and χ A commute so we can construct a basis which simultaneously diagonalizes these operators, Cpxq|C,χ y "Cpxq|C,χ y, χ A pxq|C,χ y "χ A pxq|C,χ y. However, there is a unique (up to the identification (2.13)) vacuum state -namely the one with C " 0 andχ A " x A which is invariant! We shall refer to this particular vacuum state as the QFT vacuum, | Ω QFT y " |Cpxq " 0,χ A pxq " x A y.
We next turn to the radiative sector which is spanned by r h (2)AB . Define the operator Before proceeding, let us quickly comment on the funny tensor structure in front of the integral above. We will see shortly that O AB is a graviton annihilation operator (which we will conclude from its commutator with O : AB ). Basic representation theory of the six-dimensional Poincaré group implies that this transforms in the symmetric traceless representation of the little group SOp4q. On the other hand, r h (2) transforms as a 2-tensor of GLp4q. Consequently, in order to determine the relationship between O AB and r h (2)AB , we need to understand how one can embed representations of SOp4q into those of GLp4q. This is trivial to do for any representation that doesn't explicitly rely on the invariant tensor of SOp4q, namely δ AB . For example, the vector rep. of SOp4q trivially embeds into the vector rep. of GLp4q. In the case at hand, the "traceless" symmetric rep. does depend on δ as the "tracelessness" property is defined w.r.t. δ. Consequently, any attempt to embed this into a tensor of GLp4q while preserving the symmetry structure is futile. The way to get around this is to consider instead the operator O A B " δ AC O CB which transforms in the (1,1) representation. This operator is also traceless but w.r.t. the metric δ A B which is also the invariant tensor for GLp4q! It is then clear that we can embed O A B " ş due iωu r h A (2)B and preserve the traceless property on both sides of the equation. Lowering the index on the LHS with δ and on the RHS with q as usual, we reproduce (7.4).
From (5.8), we can determine the algebra of these operators (for ω, ω 1 ą 0) where M AB,CD " 1 2 q P Q δ P tA q ButC δ DuQ`1 2 δ AtC δ DuB . The unusual tensor structure appearing in the brackets is due to the tensor structure in front of the integral in our definition (7.4). We will see that this tensor structure will be necessary to reproduce the subleading-soft theorem correctly, Though not obvious, this is precisely the algebra of momentum space creation and annihilation operators when evaluated in the QFT vacuum. To see this, we parameterize a massless momentum as we did in (2.31) It is then easy to see that (7.5) is equivalent to rO AB ppq, O : CD pp 1 qs| QFT " δ AtC δ DuB p2πq 5 p2p 0 qδ 5 p⃗ p´⃗ p 1 q.
Here, the subscript QFT indicates that the commutators have been evaluated and then simplified by evaluating them in the QFT vacuum state (7.3).
From the result above, we can immediately conclude that O AB ppq is an annihilation operator for a graviton with momentum p µ and polarization tensor ε AB given by (2.31), The annihilation operators kill all the vacuum states O AB ppq|C,χ y " 0.
Excited states are constructed by acting on the vacuum state with a creation operator, e.g. a one-graviton state with polarization tensor ε µν AB ppq is described by | p, ε ;C,χ y " O : AB ppq|C,χ y. (7.10)

Matching Conditions and Ward Identity
So far, we have restricted the discussion to the future null infinity I`. In order to discuss the scattering problem in general relativity, we also need to consider the structure of the theory on I´.
This is almost entirely identical to the analysis performed in the bulk of this paper so we will not discuss this in detail. For our purposes, it will be sufficient to note that the Hilbert space on I´is We conjecture that the boundary fields on I`and I´satisfy the following matching condition where Υ ‹ indicates the antipodal map on the celestial S 4 . 20 A proof of these matching conditions rests on the analysis of the asymptotic structure at spacelike infinity and this has only been accomplished in four-dimensional spacetimes [70][71][72][73][74]. The only discussion of this in higher dimensions that we are aware of is in [75] where the existence of BMS at spacelike infinity was studied (which is the first step in the analysis of the matching conditions). The covariant approach developed in [73,74] suggests that a similar structure likely extends to six (and all higher dimensions) in which the conditions (7.11) are realised. 21 20 The antipodal point is defined by embedding the S 4 in R 5 , x ãÑ ⃗ Xpxq and we then have ⃗ XpP pxqq "´⃗ Xpxq. For the round S 4 and stereographic embedding ⃗ Xpxq "´2 x A 1`x 2 , 1´x 2 1`x 2¯, the antipodal map is defined by P A pxq "´x A x 2 . 21 In four dimensions, to give the matching when the mass is non-vanishing, logarithmic supertranslations have to be used [74]. This is a non-trivial aspect point whose extension to higher dimensions is impossible to assess without explicit analysis. From (7.11), we find equality for the GBMS charges In the semi-classical theory, this implies a very simple Ward identity Here, S is the S-matrix operator that maps from the Hilbert space on I´to that on I`and x out |, | in y are the respective "non-interacting" multi-particle states that we are scattering.
In the rest of this section, we process (7.13) into a "soft theorem" which we can then compare to the usual soft theorem derived in standard QFT via Feynman diagrams [5,10]. To do this, we rewrite the charges T f and J Y as integrals over all of I`using the boundary conditions (2.19) for and similarly for the charges on I´.
The next step is to use the u-evolution equations (B.48) and (B.49) to "simplify" the charges.
Since we have all the relevant equations explicitly worked out, it would be possible to do this and derive the most general form of the soft theorem. However, our main goal is to derive a soft theorem that we can match to the one derived in QFT. Consequently, we shall simplify our work and consider only the special case where the in and out vacuum states are the QFT vacuum (7.3). We shall also assume that Φ 0 pxq " 0 for simplicity. 22 In this state, q AB " δ AB and h (1)AB " 0 so the evolution equations for r U (3) and Ă W (5) simplify to (B.52) which we recall here Using this, we can rewrite the charge in the form where where the derivative operator D was defined in equation (4.27) of [23], We can similarly simplify the charges on I´as well. We can then rewrite the Ward identities (7.13) as x Using the definitions (7.4) and (2.41), the soft charges can be written as We immediately see that the soft charges T S f and J S Y insert a single soft-graviton into the S-matrix so the LHS of (7.19) is the soft limit of a graviton amplitude. To simplify the RHS, we recall thatby construction -the charges T f and J Y generate GBMS transformations on the fields as in (6.12).
From this, we find where S AB is the spin-operator for the graviton. Explicitly, The precise set of terms that appear in the superrotation transformation of O AB are due to the specific tensor structure we have in front of the integral in (7.4).
Putting all of this together, we can recast the Ward identities (7.19) into the form and (7.24)

Soft Graviton Theorems
The universal factorization of a scattering amplitude with external graviton states was first studied by Weinberg in 1965 [5] who showed that In particular, this so-called leading soft-graviton theorem showcases a universal factorization of the scattering amplitude in the soft (p Ñ 0) limit. The "soft-factor" does not depend on any of the details of the theory (such as the explicit interaction terms or field content) -rather it only depends on the quantum numbers of the external states in the scattering amplitude. This universal structure plays a crucial role in resolving the infrared divergences of four-dimensional gravitational theories.
Surprisingly, it wasn't until 2014 [10] that it was finally understood that the Op1q term in (7.25) is, in fact, also universal in the same way! Including this universal piece, the complete soft limit takes the form where J i is the angular momentum operator acting on the ith one-particle state.
In this section, we show that (7.26) is equivalent to the BMS Ward identities (7.23) and (7.24).
This equivalence has already been studied in previous works [22,23,[31][32][33] but in each case, certain restrictive assumptions were made. [22] proved the equivalence of the leading soft theorem to the supertranslation Ward identity for linearised gravity. [32] generalised the result to non-linear GR but on a smaller phase space with no superrotations. Starting from the subleading soft theorem and the linear analysis of [22], [37] proposed the form of one bracket among asymptotic fields to reproduce the soft theorem. Finally, [33] working in non-linear gravity (but linearising around a Bondi frame) showed the equivalence for the subleading soft theorem but for the special case where all external states except the graviton are scalars. With all the pieces at hand, we are now able to prove the equivalence between soft theorems and BMS Ward identities in full generality.
The first step is to recast the soft-theorem into a more palatable form using the momentum parameterization (7.6). Written in terms of the operators G AB pxq and B AB pxq, the soft theorem takes the form (see [31] and [23] respectively) i ω i r´p i¨p pxqs lnr´p i¨p pxqsx out |S| in y, (7.27) and x out |B AB pxqS| in y " κ 2 where P C AB pxq " S i AB is the representation of SOpdq under which the ith particle transforms.
To prove the equivalence, we substitute (7.27) and (7.28) into the LHS of (7.23) and (7.24) respectively. We start with the leading soft theorem. We first find Then, using the property B 2ˆ1 x 2˙"´4 π 2 δ 4 pxq, (7.31) the first equation of (7.23) immediately follows. The equivalence for the subleading soft theorem similarly follows if we use the property (derived in [23]) On the other hand, the odd-dimensional case is more delicate and we can only expect that some of the conditions and ideas presented here carry over. This is essentially due to the fact that Huygens' principle applies in even dimensions, but not in odd dimensions. More precisely, in even dimensions, Green's functions for massless particles are localized on the light-cone implying a simple and local asymptotic structure. In odd dimensions, Green's functions are supported on the interior of the lightcone which leads to various non-localities on I˘. This was explored for gauge theories in [26][27][28] but the analogous structure for gravity is yet to be understood. Given the results of [23,31], it is reasonable to conjecture that the local structure in even dimensions and the non-local structure in odd dimensions can be treated in a unified manner by utilizing the shadow transform. We leave an exploration of this for future work.
It would also be of utmost interest to analyse the consequences of the phase space prescriptions put forward in this paper in four dimensions. In order to comment on this while concluding, let us summarise our results.
We started from the general solution space derived in [30] and we have restricted it by requiring flatness at the past boundary of null infinity. Such conditions generate analytic expansions in powers of r´1. We have studied both large r and large u divergences appearing in the pre-symplectic potential θ and symplectic form Ω. The large r divergences were renormalised by adding local and covariant counterterms in θ and the explicit large u were removed by requiring that Ω be non-degenerate. The non-degeneracy requirement removes the asymptotic Weyl rescalings and reduces the ASG from WBMS transformations to the GBMS transformations, which consist only of supertranslations and superrotations. We then constructed the Poisson brackets from the symplectic form so obtained.
We also found from this that the symplectic form for GBMS transformations is non-integrable in field space, namely Ωpδ, L ξ q ‰ δH ξ and hence there is no unambiguous notion of finite charges. In other words, GBMS diffeomorphisms are not canonical. This result is expected from previous works on four-dimensional gravity [43,50,65]. It is, however, possible to extract the integrable part of the symplectic form and define charges which faithfully represent the GBMS algebra. Such charges act consistently on the solution space but their action on the fields is not the Lie derivative L ξ .
The renormalization of the large r divergences with local and covariant counterterms and our treatment of large u divergences in higher dimensions appear to be novel. It also seems that our analysis will imply something new for four-dimensional gravity as well. Previous discussions of renormalization do not discuss covariantisation of the counterterms and remain vague about the large u behaviour and invertibility [43][44][45] (see however also [76]) . It is important to check that the analysis performed in this paper applies in a straightforward way to four dimensions as well and we leave this exploration for future work. Let us highlight some points in this regard: • The addition of local and covariant counterterms at diverging orders also modifies the finite part of the symplectic form and hence affects the parts we identify as integrable and nonintegrable. The resulting split of δH ξ into integrable rδH ξ s int and non-integrable rδH ξ s non-int parts appears to be less ambiguous because it is obtained from this sort of first-principle prescription, although ambiguities still remain if we insist that we can shift δH ξ as This equation is equivalent to saying that we can split Ωpδ, L ξ q into integrable and nonintegrable parts. In previous literature [65] F -and hence the integrable part of the symplectic form -was selected by imposing that it is zero in the absence of news [43,44,65,77]. In the case in which only supertranslations are allowed, F is related to the passage of gravitational waves through null infinity and the condition acquires a clear physical meaning.
• We obtained the Poisson structure by inverting the gravitational symplectic form obtained with covariant phase space methods. Given that the finite charges are obtained only on the basis of the integrable part of Ω, L ξ is not the Hamiltonian vector field associated to such charges. Future analysis could study the algebraic structures stemming from the brackets proposed by Barnich and Troessaert [50], which in the case of (8.2) read as Such brackets are not naively related to a symplectic form [78], but we stress that many of our conclusions -for example the removal of the Weyl freedom -are highly motivated by the systematic analyisis of the symplectic form output by the covariant phase space formalism.
In reflecting on these matters, it would be interesting to include also the logarithmic terms identified in [30] (briefly discussed here in Section 2.5) both in four and higher even dimensions in order to have a more detailed and general understanding of asymptotically flat gravitational phase spaces and their holographic properties.

Acknowledgements
We

A Ricci Tensor Components in Bondi-Sachs Gauge
In this appendix, we work out the Riemann tensor in Bondi-Sachs gauge in general D " d`2 dimensions. The six-dimensional formulas are obtained by setting d " 4.

Metric and Inverse
The line element takes the form The Bondi-Sachs gauge condition implies G rr " G rA " G uu " G uA " 0. The metric components and its inverse are given by and whereh is the matrix inverse of h. The root determinant of the metric is b´d etpG µν q " r d e 2β ? q. (A.5)

Christoffel Symbols
The Christoffel symbol is defined by Γ λ µν rGs " The components of the Christoffel symbol then work out to be Γ u rr rGs " 0, Γ u rA rGs " 0, Γ u AB rGs " and Γ r uu rGs " Γ r rr rGs " 2B r β,

Ricci Tensor
The Ricci tensor is given by As in the rest of the paper, we have also employed matrix notation in the formulas above. Repeated indices are contracted w.r.t the metric q AB and D A is the covariant derivative w.r.t. q.
Next, we have Here, R AB is the Ricci tensor w.r.t. q AB . From this, we can deduce that (A. 16) Finally, we have and R uA rGs " (A.18)

B Asymptotic Expansions
In this Appendix, we describe the large expansions of solutions in S general , S analytic and S canonical .
I. R rr rGs " 0: This implies This equation fixes the large r components of β, II: R rA rGs " 0: This implieś B r pr 6 e´2 β hB r W q "´2r 8 B r pr´4Dβq`r 4 D¨phB r hq`1 2 r 4 trrB rh Dhs. (B.7) The RHS of this equation has the form RHS " r 2 W (2)`r rW (3)`W(3, 1) ln rs`rW (4)`W(4, 1) ln rs W (5, 1)`W(5, 2) ln r`W (5, 3) ln 2 r r`O pr´2q. (B.8) Using the explicit forms of the large r expansions, we find W (2) "´D¨h (1) , We can solve for the large r coefficients of W as Using this, we can fix the large r coefficients for U , Note that U (3) is not fixed by this equation.
IV: R AB rGs " 0: This implieś where We can determine the following evolution equations for the large r coefficients of h Note that there is no evolution equation for h (2)AB .
Step V: R uu rGs " 0: This implies an evolution equation for U (3) ,

(B.17)
Step VI: R uA " 0: This implies an evolution equation for W (5) but we will not need this equation here.

B.2 Analytic Solutions
In this section, we describe the large r expansion for analytic solutions. These are obtained from S general by imposing an additional constraint r 4 e μ µ e ν ν e ρ ρ e σ σ R µνρσ rGs| I`" 0.
We start by proving that this constraint does, in fact, remove all the log terms in the large r expansion. To process the constraint, we need to first introduce a relevant set of vielbein. We Here, E a A is the normalized vielbein for h AB so that h AB " δ ab E a A E b B and E A a " h AB δ ab E b B . With this choice, g µν "´2n pµ ℓ νq`δab e µ a e ν b .
1. R ABCD rGs| I`" 0 at Opr 2 q: At this order, we find the constraint C ABCD rqs " 0. This implies that q AB is conformally flat so we can write q AB pxq " e 2Φpxqq AB pxq,q AB pxq " δ CD B A χ C pxqB B χ D pxq. imply that all the log terms in the large r expansion vanish identically!

Explicit Large r Expansion
Having shown that all the log terms vanish once the Riemann tensor constraint has been imposed, we now present explicit formulas for all the large r coefficients of the metric.
The large r coefficients of h AB satisfy the following evolution equations B u r h (1)AB "´R tABu , B u r h (3)AB " 2 3 D tA pD¨r h (2) q Bu´1 2 D 2 r h (2)AB`p R r h (2) q tABu´1 12 R r h (2)AB ,   It was argued, correctly, that the additional part with respect toN AB carries the information of the "Geroch tensor", but the identification with the true degrees of freedom in the symplectic form was not made. Indeed, the correctly gauge-invariant quantity was later found in [32]  C Finite BMS Transformations from S canonical Ñ S analytic : Details In this Appendix, we present the details for the finite BMS transformation which maps solutions from S canonical to those in S analytic . Given a metricG in S canonical , the new metric G obtained via a diffeomorphism is given by G µν "´e 2βpXqŮ pXqB µů B νů´e 2βpXq rB µů B νr`Bµr B νů sg AB pXq " B µx A´W A pXqB µů ı " B νx B´W B pXqB νů ı . (C.1) We are interested in diffeomorphisms which preserve the Bondi-Sachs gauge conditions, G rr " G rA " B r detpr´2G AB q " 0. (C.2) We start by analysing the equations (C.1) at leading order in r whereX µ pXq has the form u "ů (0)`O pr´1q,r " rr p´1q`O p1q,X A "X A (0)`O pr´1q. (C.3) Substituting this into (C.1) and expanding to leading order, we find G uu " r 2 rB uX(0) s 2`O prq, G ur "´B uů(0)rp´1q`O pr´1q.

(C.4)
Since we need G uu " Op1q and G ur "´1`Opr´2q we must haveX A (0) " χ A pxq, B uů(0)rp´1q " 1. Using this and moving to subleading order, we find It follows from this that X A (0) " χ A pxq,ů (0) " κpu, xq " e´Φ pxq u`Cpxq,r (1) " e Φpxq . (C.6) Having completed the leading order analysis, we move on to the full large r expansion ofX µ pXq which takes the form u " κ`e´Φů (1) r`e´2 Φů ( (C.7) Substituting this expansion into (C.1) and imposing the conditions (C.2), we can obtain each of the coefficients shown above. The general procedure is as follows. We first analyse G rA " 0 at Opr´nq which fixesx A (n`1) " B χ B x AxB (n`1) . We next analyse G rr " 0 at Opr´n´2q which fixesů (n`1) . We then analyse the final constraint B r detpG AB {r 2 q " 0 at Opr´n´2q to fixr (n) . This procedure is completed iteratively for each n (starting at n " 0).
Here, we shall simply present the results of this analysis. It will be convenient to introduce the following quantities